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Properties

Label 2-168-8.5-c1-0-11
Degree $2$
Conductor $168$
Sign $-0.320 + 0.947i$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.835 − 1.14i)2-s − i·3-s + (−0.602 − 1.90i)4-s − 0.467i·5-s + (−1.14 − 0.835i)6-s + 7-s + (−2.67 − 0.907i)8-s − 9-s + (−0.532 − 0.390i)10-s + 4.87i·11-s + (−1.90 + 0.602i)12-s − 4.56i·13-s + (0.835 − 1.14i)14-s − 0.467·15-s + (−3.27 + 2.29i)16-s + 6.09·17-s + ⋯
L(s)  = 1  + (0.591 − 0.806i)2-s − 0.577i·3-s + (−0.301 − 0.953i)4-s − 0.208i·5-s + (−0.465 − 0.341i)6-s + 0.377·7-s + (−0.947 − 0.320i)8-s − 0.333·9-s + (−0.168 − 0.123i)10-s + 1.47i·11-s + (−0.550 + 0.173i)12-s − 1.26i·13-s + (0.223 − 0.304i)14-s − 0.120·15-s + (−0.818 + 0.574i)16-s + 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.320 + 0.947i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.320 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.860860 - 1.20036i\)
\(L(\frac12)\) \(\approx\) \(0.860860 - 1.20036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.835 + 1.14i)T \)
3 \( 1 + iT \)
7 \( 1 - T \)
good5 \( 1 + 0.467iT - 5T^{2} \)
11 \( 1 - 4.87iT - 11T^{2} \)
13 \( 1 + 4.56iT - 13T^{2} \)
17 \( 1 - 6.09T + 17T^{2} \)
19 \( 1 - 1.34iT - 19T^{2} \)
23 \( 1 + 4.09T + 23T^{2} \)
29 \( 1 - 7.78iT - 29T^{2} \)
31 \( 1 - 4.40T + 31T^{2} \)
37 \( 1 + 4.40iT - 37T^{2} \)
41 \( 1 + 6.09T + 41T^{2} \)
43 \( 1 - 4.15iT - 43T^{2} \)
47 \( 1 + 6.68T + 47T^{2} \)
53 \( 1 + 1.34iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 5.49iT - 61T^{2} \)
67 \( 1 - 5.90iT - 67T^{2} \)
71 \( 1 + 4.72T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 - 7.96T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.47788254571172121998864106876, −11.88157400192405441845827859012, −10.51022010514172922793264523164, −9.844776344551184483355269367950, −8.386554226050920696695834441133, −7.22468779232690582674775652024, −5.73489704740383284687347256253, −4.75333451185973207602629050827, −3.11707783232254985411901279214, −1.49712371513229622638503189680, 3.13987186324223988705766447773, 4.34355661065666302194494839715, 5.57768603612114509129243561553, 6.55741893326754876522924223990, 7.969707466251955478597547843137, 8.764852980239208181445869786986, 10.00402837825757457631101535350, 11.40826073031971527922728180527, 11.98183181851514442091582132190, 13.54385015455499181043948231398

Graph of the $Z$-function along the critical line